The first summand, given in (REF ) in terms of \(U(x):=\frac{1+2 \cos (2\pi x)}{3},\)  we refer to as the anti-chiral part, and it describes tunneling between A-A\(^{\prime }\) /B-B\(^{\prime }\)  atoms. The second summand defined by \(U^{\pm }_{\operatorname{c}}(x):=\tfrac{1- \cos (2\pi x)\pm \sqrt{3} \sin (2\pi x)}{3} = \tfrac{1 \pm 2\sin (2\pi x\mp \pi /6)}{3}, \)  is the
chiral part modeling the tunneling between A-B\(^{\prime }\) /B-A\(^{\prime }\)  atoms. Here, A and B correspond to the two different representatives of the fundamental cell of a honeycomb lattice and \(^{\prime }\)  indicates atoms of the second lattice. This terminology is inspired by the Bistritzer-MacDonald model [1]}.
The Hamiltonian \(H_{w}(\theta ,\phi ,\vartheta ):\ell ^2({\mathbb {Z}};{\mathbb {C}}^4) \rightarrow \ell ^2({\mathbb {Z}};{\mathbb {C}}^4)\)  is then, for some fixed length \(L>0\)  of the moiré cell, given by
\( H_{w}(\theta ,\phi ,\vartheta )\psi _n = D_{\operatorname{kin}}(\theta )\psi _n + V_w(\vartheta + \tfrac{n}{L},\phi )\psi _n\) 
